Russia has promised free accommodation to residents of Ukraine’s partially occupied Kherson region who want to evacuate in the face of continuing and sustained Ukrainian military gains along the southern front.

Russian deputy prime minister Marat Khusnullin made the announcement shortly after the Russia-backed leader of Kherson, one of four Ukrainian regions illegally annexed by Moscow last month, asked the Kremlin to organise an evacuation from four cities in the region.

Vladimir Saldo said in a video posted online: “Cities of the Kherson region – Kherson and Nova Kakhovka, Hola Prystan and Chornobaivka – are subject to daily missile strikes.

“These missile strikes cause serious damage, first and foremost to the residents. Among targets missiles hit are hotels, residential buildings, markets – (places) where there are lots of civilians.”

Mr Saldo said a decision has been made to evacuate Kherson residents to the Russian regions of Rostov, Krasnodar and Stavropol, as well as the annexed Crimea.

“I would like to ask you to help organise this process,” he said. “We, residents of the Kherson region, of course know that Russia doesn’t abandon their own, and Russia always offers a hand.”

His plea comes as Ukrainian forces push their counter-offensive deeper into the southern Kherson region, albeit at a slower pace.

Ukrainian armed forces have reported steady territorial gains along the southern front, including the recapture of 75 settlements in the Kherson region in the last month, the Ministry for Reintegration of Temporarily Occupied Territories said on Thursday night.

In the east, Ukraine’s armed forces have recaptured 502 settlements in the Kharkiv region, 43 in the Donetsk region and seven in the Luhansk region, the ministry said.

Mr Saldo’s deputy, Kirill Stremousov, tried to play down the announcement, saying that “no-one’s retreating … no-one is planning to leave the territory of the Kherson region”.

Earlier on Thursday, the British military said on Twitter that “the Russian occupation authorities have likely ordered preparation for the evacuation of some civilians from Kherson”.

The Ministry of Defence said: “It is likely that they anticipate combat extending to the city of Kherson itself.”

Early on Friday, Russia continued its targeted attacks on critical infrastructure across Ukraine, a strategy that started on Monday when a massive, coordinated attack on nearly every region in the country was said to be carried out in retaliation for the explosion on a Moscow-funded bridge linking Crimea to the Russian mainland.

Multiple Russian missile strikes shook the Zaphorizhzhia region capital overnight as the city continued to be a focal point while Ukraine pushed its counter-offensive on the southern front.

Zaporizhzhia regional Governor Oleksandr Starukh said several explosions were reported in the city overnight at infrastructure facilities, causing fires.

There were no victims in preliminary reports, and further details about specific damage were unavailable. Russian forces have struck the regional capital and surrounding area continuously in recent days and weeks, creating concerns about the safety of the nearby nuclear power plant.

The regional capital is about 100 miles from the plant, the largest nuclear power facility in Europe. Two days ago, it was forced to revert to diesel-fuelled generator power to maintain its reactor cooling systems after an attack on a substation’s communication line was lost during fighting in the area.

Missile, drone and rocket attacks on Ukraine have kept the country on edge with air raid sirens occurring more frequently and bringing a heightened sense of urgency after Monday’s strike killed 19 and wounded more than 100, including many in the capital, Kyiv.